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Leading Order Response of Statistical Averages of a Dynamical System to Small Stochastic Perturbations

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Abstract

The classical fluctuation-dissipation theorem predicts the average response of a dynamical system to an external deterministic perturbation via time-lagged statistical correlation functions of the corresponding unperturbed system. In this work we develop a fluctuation-response theory and test a computational framework for the leading order response of statistical averages of a deterministic or stochastic dynamical system to an external stochastic perturbation. In the case of a stochastic unperturbed dynamical system, we compute the leading order fluctuation-response formulas for two different cases: when the existing stochastic term is perturbed, and when a new, statistically independent, stochastic perturbation is introduced. We numerically investigate the effectiveness of the new response formulas for an appropriately rescaled Lorenz 96 system, in both the deterministic and stochastic unperturbed dynamical regimes.

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Notes

  1. The author privately communicated that his response formula applies to a perturbation noise in the Stratonovich form.

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Acknowledgements

The work was supported by the Office of Naval Research Grant N00014-15-1-2036.

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  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan st., Chicago, IL, 60607, USA

    Rafail V. Abramov

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  1. Rafail V. Abramov
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Correspondence to Rafail V. Abramov.

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Abramov, R.V. Leading Order Response of Statistical Averages of a Dynamical System to Small Stochastic Perturbations. J Stat Phys 166, 1483–1508 (2017). https://doi.org/10.1007/s10955-017-1721-2

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